Sets of Minimality of (1 − 1)-Rational Functions

We describe dynamical systems associated to (1 ? 1)-rational functions on the field of p-adic numbers.We focus on sets of minimality of such systems.     

Commutator identities of the homotopes of (−1, 1)-algebras

We study the commutator algebras of the homotopes of (?1, 1)-algebras and prove that they are Malcev algebras satisfying the Filippov identity h _a (x, y, z) = 0 in the case of strictly (?1, 1)-algebras. We also proved that every Malcev algebra with the identities xy ³ = 0, xy ² z ² = 0, and h _a (x, y, z) = 0 is nilpotent of index at most 6.     

A simple proof of Ramanujan’s summation of the1ψ1

A simple proof by functional equations is given for Ramanujan’s₁ ψ ₁ sum. Ramanujan’s sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series.     

Structures of Wey1 Groups of Some Kac-Moody Algebras

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The localization of 1-cohomology of transitive Lie algebroids

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.     

An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors

Let $\text{Ω}$ be a bounded open connected subset of $\text{R}{}^{\mathrm{n}}$ with a Lipschitz-continuous boundary and let $\text{Θ}\text{∈}\text{C}{}^1\text{(}\text{Ω}\text{;}\text{R}{}^{\mathrm{n}}\text{)}$ be a deformation of the set $\text{Ω}$ satisfying $\text{det}\text{?Θ}\text{>0}$ in $\text{Ω}$. It is established that there exists a constant $\text{C(}\text{Θ}\text{)}$ with the following property: for each deformation $\text{Φ}\text{∈H}{}^1\text{(Ω;}\text{R}{}^{\mathrm{n}}\text{)}$ satisfying $\text{det}\text{?Φ}\text{>0}$ a.e. in $\text{Ω}$, there exist an n×n rotation matrix $\text{R}\text{=}\text{R}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ and a vector $\text{b}\text{=}\text{b}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ in $\text{R}{}^{\mathrm{n}}$ such that

The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Müller. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The logic of π1-conservativity

We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic of ₁-conservativity over eachbE ₁-sound axiomatized theory containingI ₁ (PA with induction restricted tobE ₁-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a separate continuation of the present paper).     

On the structure of Δ 1 4 -sets of reals

Assuming that an inaccessible cardinal exists, we construct a ZFC-model where every ₁ ⁴ -set is measurable but there exists a ₁ ⁴ -set without the property of Baire. By a result of Shelah, an inaccessible cardinal is necessary for this result.The author is partially supported by the Basic Research Foundation of the Israel Academy of Sciences and by the Swiss National Funds.     

On the Convergence of ω1 Sequences of Real Functions

We study sets of points at which ω₁ sequences of real functions from a given class F converge. As F we consider continuous functions, first class of Baire, Borel measurable functions, functions with Baire property and Lebesgue measurable functions. Connections of those problem with additional set-theoretic axioms are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields

In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ?-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ?-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ?-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.     

Values of Minors of (1, −1) Incidence Matrices of SBIBDs and Their Application to the Growth Problem

We obtain explicit formulae for the values of the v-j minors, j=0,1,2 of (1, –1) incidence matrices of SBIBD(v,k,). This allows us to obtain explicit information on the growth problem for families of matrices with moderate growth. An open problem remains to establish whether the (1, –1) CP incidence matrices of SBIBD(v,k,), can have growth greater than v for families other than Hadamard families.     

On Maximal Abelian Self-adjoint Subalgebras of Factors of Type Ⅱ1

In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.     

Convergence acceleration of continued fractions of Poincaré's type 1

A new method of convergence acceleration is proposed for continued fractions of Poincaré's type 1. Each step of the method (and not only the first one, as in the Hautot method [1]) is based on an asymptotic behaviour of continued fraction tails. A theorem is proved detailing properties of the method in six cases considered here. Results of numerical tests for all Hautot's examples confirm a good performance of the method.     

Chern Classes of Gauss–Manin Bundles of Weight 1 Vanish

We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at dies up to torsion in the Chow ring, except in codimension 0.     

On Galois representations arising from towers of coverings of P1\{0, 1, ∞}

Summary We propose a new way to describe, universally, thel-adic Galois representations associated to each almost pro-l tower of etale coverings ofP ¹\{0, 1, }. This generalizes our universal power series for Jacobi sums (cf. [I]) which arises from the tower of Fermat curves of degreel ⁿ (n), and contains the case of the tower of modular curves of level 2ml ⁿ (m: fixed,n) as another important special case. As a fundamental tool, we shall establish and use an almost pro-l version of the theorems of Blanchfield and of Lyndon in Fox free differential calculus.     

Facet of regular 0–1 polytopes

The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. Some connections are established between prime implicants of a monotone or a regular Boolean function on the one hand, and facets of the convex hullH of the zeros of on the other. In particular (Corollary 2) a necessary and sufficient condition is given for a constraint of a covering problem to be a facet of the corresponding integer polyhedron. For any prime implicantP of, a nonempty familyF(P) of facets ofH is constructed. Proposition 17 gives easy-to-determine sharp upper bounds for the coefficients of these facets when is regular. A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet ofH, and this facet has 0–1 coefficients. Every nontrivial facet ofH with 0–1 coefficients is obtained from this class.